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# Multivalued Dependencies - PowerPoint PPT Presentation

Multivalued Dependencies. Fourth Normal Form. Sources: Slides by Jeffrey Ullman book by Ramakrishnan & Gehrke. Limitations of FD’s. Some redundancies cannot be detected using just functional dependencies

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### Multivalued Dependencies

Fourth Normal Form

Sources: Slides by Jeffrey Ullman

book by Ramakrishnan & Gehrke

• Some redundancies cannot be detected using just functional dependencies

• Example: suppose a teacher can teach several courses, several books can be recommended for a course, and same book can be recommended for different courses.

• Suppose all this information is smushed together into one relation CTB

• There are no FD’s, so key is CTB and relation is in BCNF

• But there is redundancy since course implies book

• Eliminate redundancy by decomposing into CT and CB

• Notion of MVD captures redundancy that FD’s can’t

• A multivalued dependency (MVD) on R, X ->->Y, says that if two tuples of R agree on all the attributes of X, then their components in Y may be swapped, and the result will be two tuples that are also in the relation.

• i.e., for each value of X, the values of Y are independent of the values of R-X-Y.

• A consumer’s phones are independent of the candies they like.

• name->->phones and name ->->candiesLiked.

• Thus, each of a consumer’s phones appears with each of the candies they like in all combinations.

• This repetition is unlike FD redundancy.

• name->addr is the only FD.

sue a p1 b2

Then these tuples must also be in the relation.

Tuples Implied by name->->phones

If we have tuples:

sue a p1 b1

sue a p2 b2

Picture of MVD X ->->Y

XY others

equal

exchange

• Every FD is an MVD (promotion ).

• If X ->Y, then swapping Y ’s between two tuples that agree on X doesn’t change the tuples.

• Therefore, the “new” tuples are surely in the relation, and we know X ->->Y.

• Complementation : If X ->->Y, and Z is all the other attributes, then X ->->Z.

• Like FD’s, we cannot generally split the left side of an MVD.

• But unlike FD’s, we cannot split the right side either --- sometimes you have to leave several attributes on the right side.

Consumers(name, areaCode, phone, candiesLiked, manf)

• A consumer can have several phones, with the number divided between areaCode and phone (last 7 digits).

• A consumer can like several candies, each with its own manufacturer.

• Since the areaCode-phone combinations for a consumer are independent of the candiesLiked-manf combinations, we expect that the following MVD’s hold:

name ->-> areaCode phone

name ->-> candiesLiked manf

Here is possible data satisfying these MVD’s:

name areaCode phone candiesLiked manf

Sue 650 555-1111 Twizzlers Hershey

Sue 650 555-1111 Smarties Nestle

Sue 415 555-9999 Twizzlers Hershey

Sue 415 555-9999 Smarties Nestle

But we cannot swap area codes or phones by themselves.

That is, neither name->->areaCode nor name->->phone

holds for this relation.

• The redundancy that comes from MVD’s is not removable by putting the database schema in BCNF.

• There is a stronger normal form, called 4NF, that (intuitively) treats MVD’s as FD’s when it comes to decomposition, but not when determining keys of the relation.

• A relation R is in 4NF if: whenever X ->->Y is a nontrivial MVD, then X is a superkey.

• NontrivialMVD means that:

• Y is not a subset of X, and

• X and Y are not, together, all the attributes.

• Note that the definition of “superkey” still depends on FD’s only.

• Remember that every FD X ->Y is also an MVD, X ->->Y.

• Thus, if R is in 4NF, it is certainly in BCNF.

• Because any BCNF violation is a 4NF violation (after conversion to an MVD).

• But R could be in BCNF and not 4NF, because MVD’s are “invisible” to BCNF.

• If X ->->Y is a 4NF violation for relation R, we can decompose R using the same technique as for BCNF.

• XY is one of the decomposed relations.

• All but Y – X is the other.

MVD’s: name ->-> phones

name ->-> candiesLiked

• Key is {name, phones, candiesLiked}.

• All dependencies violate 4NF.

• Decompose using name -> addr:

• In 4NF; only dependency is name -> addr.

Consumers2(name, phones, candiesLiked)

• Not in 4NF. MVD’s name ->-> phones and name ->-> candiesLiked apply. No FD’s, so all three attributes form the key.

(Sadly, no simple rule for projecting MVD’s onto decomposed relations – use heuristics and knowledge of application)

• Either MVD name ->-> phones or name ->-> candiesLiked tells us to decompose to:

• Consumers3(name, phones)

• Consumers4(name, candiesLiked)

• 4NF  BCNF  3NF